| L(s) = 1 | − 18·3-s + 10·5-s − 138·7-s + 243·9-s − 544·11-s + 338·13-s − 180·15-s + 1.22e3·17-s + 522·19-s + 2.48e3·21-s + 1.63e3·23-s − 2.93e3·25-s − 2.91e3·27-s + 2.20e3·29-s − 874·31-s + 9.79e3·33-s − 1.38e3·35-s − 2.16e3·37-s − 6.08e3·39-s − 1.26e4·41-s + 1.77e4·43-s + 2.43e3·45-s − 4.30e3·47-s − 1.60e4·49-s − 2.21e4·51-s − 1.20e4·53-s − 5.44e3·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.178·5-s − 1.06·7-s + 9-s − 1.35·11-s + 0.554·13-s − 0.206·15-s + 1.03·17-s + 0.331·19-s + 1.22·21-s + 0.643·23-s − 0.938·25-s − 0.769·27-s + 0.487·29-s − 0.163·31-s + 1.56·33-s − 0.190·35-s − 0.259·37-s − 0.640·39-s − 1.17·41-s + 1.46·43-s + 0.178·45-s − 0.283·47-s − 0.957·49-s − 1.18·51-s − 0.587·53-s − 0.242·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 p T + 3034 T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 138 T + 35134 T^{2} + 138 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 544 T + 188662 T^{2} + 544 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1228 T + 1920310 T^{2} - 1228 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 522 T + 2294638 T^{2} - 522 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1632 T + 6071278 T^{2} - 1632 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2208 T + 28123318 T^{2} - 2208 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 874 T + 53919822 T^{2} + 874 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2160 T + 136107718 T^{2} + 2160 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12638 T + 179075962 T^{2} + 12638 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 17760 T + 285701350 T^{2} - 17760 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4300 T - 81499586 T^{2} + 4300 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12020 T + 871681390 T^{2} + 12020 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 27532 T + 1410254998 T^{2} - 27532 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 38016 T + 747757270 T^{2} + 38016 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 90974 T + 4517034702 T^{2} - 90974 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4384 T + 2552860222 T^{2} - 4384 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 95312 T + 6403119726 T^{2} + 95312 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 62168 T + 6089378718 T^{2} - 62168 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 137892 T + 123442466 p T^{2} + 137892 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 142534 T + 14051880898 T^{2} - 142534 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4896 T + 16283966302 T^{2} - 4896 p^{5} T^{3} + p^{10} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.678958689609252724537369260323, −9.477815504490941115121214664546, −8.725771934301218045268133338036, −8.304192338778515973801667241887, −7.60724587764878617993143880896, −7.50954560898806694613235451270, −6.82231759500856975565150301213, −6.33992315426398602797732116656, −6.01585388388646832547904911472, −5.58311219603766841889249629009, −4.98728397627757492445678454710, −4.88074614564744608407083528795, −3.83493834164491573534160081366, −3.53821456161442806116946736494, −2.88946895464659646661669852771, −2.32828038171805711286353453077, −1.41501269918836471923697417949, −0.958510028451292478221083739894, 0, 0,
0.958510028451292478221083739894, 1.41501269918836471923697417949, 2.32828038171805711286353453077, 2.88946895464659646661669852771, 3.53821456161442806116946736494, 3.83493834164491573534160081366, 4.88074614564744608407083528795, 4.98728397627757492445678454710, 5.58311219603766841889249629009, 6.01585388388646832547904911472, 6.33992315426398602797732116656, 6.82231759500856975565150301213, 7.50954560898806694613235451270, 7.60724587764878617993143880896, 8.304192338778515973801667241887, 8.725771934301218045268133338036, 9.477815504490941115121214664546, 9.678958689609252724537369260323
